Connected and Path Connected

Which of the following subsets of are a). path connected b). connected:

i).

ii).

iii).

I think that:

i). I don't think this is path connected. If you took a point in each set , for example and , you couldn't get from one to the other in a continuous path. This is because you would have to travel through 0, but 0 isn't in either set.

ii). I think this is path connected. Since both sets now contain 0, you could travel between them.
Since they are path connected, the subsets are connected.

iii). I think this is path connected. One set contains 0, the other contains points arbitrarily close to 0. Therefore you could make a continuous map between the two.
Once again, since they are path connected they are connected.

P.S: Thanks to everyone who has helped me with any topology questions over the past few days. You've helped me so much!

You (should) know that balls are connected, so (2) is two connected spaces with nonempty intersection, and (3) is two connected spaces each of which intersects a third connected space (think about the path you'd like to draw). There are theorems you can cite.

Edit: I wasn't as clear as I could have been on (3). The reason this one isn't as straightforward as (2) is that the two balls have empty intersection. But onsider the set , which is a straight-line path from the center of one ball to the center of the other. If you call the two balls , then you can write , and then apply the theorem about connected spaces with nonempty intersection.